Spring 2015

In a certain sense, knowledge of all rotation numbers together with certain homological data determines a representation of a group in Homeo+(S^1), up to the relation of semi-conjugacy. With that goal in mind, we can ask Given a free group F , and an element w of F, and given values of the rotation numbers of the generators, what is the
set of possible rotation numbers of w? In this talk, we show that there are strong constraints on the dynamics of free group actions on the circle that maximize the rotation number, and the constraints are powerful enough to guarantee rational rotation numbers. We also show that the graphs of extremal rotation numbers associated to positive words in free groups (aka Ziggurats) satisfy interesting stability and self-similarity conditions.

April 9, 2015

A Panorama of Extremal Length Geometry. — Ian Frankel

The extremal length of a curve (or, more generally, a
multicurve, or, more generally, a measured foliation) in a hyperbolic
Riemann surface is an important conformal invariant. We will relate
extremal and hyperbolic lengths, use extremal lengths to characterize the
Teichmueller metric, and explain why Teichmueller space is not negatively
curved.

April 16, 2015

Sections of surface bundles and Diophantine equations — Nick Salter

André Weil’s “three pillars” philosophy instructs us to look for a dictionary between notions in number theory and in Riemann surface theory. In the arithmetic world, Diophantine equations play a fundamental role. In this talk, I will explain why surface bundles over surfaces are a topological counterpart to Diophantine equations, and why sections of surface bundles over surfaces play the role ofsolutions to Diophantine equations. I will discuss some of the places where these two theories run parallel to each other, and some other aspects which are quite divergent.

April 23, 2015

Circle actions on the boundary of Schottky space — Alden Walker

To a complex parameter c, we associate the two-generator iterated function system {z -> cz-1, z -> cz+1}. I’ll describe how the IFS for certain parameters (those on the boundary of the connectedness locus) can give rise to circle actions. A finite amount of data encoded in these circle actions describes the set of cut points in the limit set of the IFS. In addition, these circle actions can be thought of as double covers of Lorenz maps and generalizations. This talk should be broadly accessible, and pictures will be provided. This is joint work with Danny Calegari, building on previous work with Danny Calegari and Sarah Koch.

What if we want to ask similar questions about multivariable polynomials? For example, how many smooth hypersurfaces of degree d in P^n are there over a finite field F_q?

In this talk we will present three seemingly different results about randomness in a finitely generated nilpotent group: an asymptotic shape theorem for First Passage Percolation (FPP); a generalization to random metrics of Pansu’s theorem that the unique asymptotic cone of a nilpotent group is a particularly nice nilpotent Lie group; a Subadditive Ergodic Theorem for nilpotent groups. The results are all related, and the proof involves sub-Riemannian geometry and Ergodic Theory.

The Borel conjecture predicts that every finitely presented Poincare duality group is the fundamental group of a unique closed aspherical manifold. I will discuss some basic things about this conjecture together with (i) some recent work of Bartels-Lueck-Weinberger which solves Borel’s conjecture for a class of hyperbolic groups, and (ii) a relative version of (i) which is joint work with J. Lafont.

May 14, 2015

Knots, polynomials, and primes — Weiyan Chen

This talk will be about a very classical object (defined in 1923) which everyone should know (in my opinion): the Alexander polynomial. It is a magic polynomial which on the one hand encodes lots of information about the knot, and on the other hand, is highly computable. I will demonstrate at least two ways to compute Alexander polynomials. If time permits, I will also explain some examples which suggest that knots and primes are analogous objects. The talk will contain lots of examples/pictures and no proofs.

May 21, 2015

Homology of covers of finite graphs — Sebastian Hensel

In the 1930s Chevalley and Weil gave a formula to describe the

homology of a finite regular cover of a Riemann surface as a representation

of the deck group. A completely analogous formula exists for covers of

finite graphs.

In this talk we will describe the beginnings of a dictionary between

topological and representation-theoretic properties of the homology

of a finite cover of a graph. This point of view allows on the one hand to

Out(F_n)) and on the other hand suggests a rich class of unsolved problems.

Everything new in this talk is joint work with Benson Farb.

May 28, 2015

Higher rank rigidity in projective geometry — Wouter van Limbeek

Locally symmetric spaces (e.g. hyperbolic manifolds) form a class of manifolds exhibiting amazing rigidity from both algebraic and geometric points of view, especially when they have so-called “higher rank”. They arise as quotients of bounded convex domains in either a projective space or a Grassmannian by a group of projective transformations. Kac-Vinberg have shown there is no rigidity in the “rank one” setting, namely they construct nonsymmetric domains in projective spaces that nevertheless admit compact quotients. On the other hand we establish rigidity in the case of “highest possible rank”: we prove that in the Grassmannian of p-planes in every bounded convex domain with a compact quotient is symmetric. This is joint work with Andrew Zimmer.

Is complete parabolicity of a translation surface (i.e. if you see one cylinder in a given direction, then that direction is covered by parallel cylinders whose moduli have rational ratios) equivalent to the surface being stabilized by a lattice in SL(2,R)?

We will completely answer these questions and relate questions about Teichmuller curves to representation theory and (linear) algebraic geometry. This talk will be based on a recent preprint by Wright, Lanneau, and Nguyen.

The talk will be self-contained, i.e. no prior knowledge of translation surfaces, Veech surfaces, Veech groups, etc. is expected.